Great IdeasNature and science

Shape, space and proof

Points, lines and figures, and proving things about them.

The question

What can we prove about shapes and space? Is Euclid's geometry the true geometry of the world?

The stops on a time line 8 stops, from 385 BC to 1916. Each number is a stop below, placed at its year.
  1. 1 Plato 385 BC
  2. 2 Plato 360 BC
  3. 3 Euclid 300 BC
  4. 4 Euclid 300 BC
  5. 5 Archimedes 250 BC
  6. 6 René Descartes 1637
  7. 7 Henri Poincaré 1902
  8. 8 Albert Einstein 1916

How the books answer

Oldest first. Each stop is our own short summary of what the author argued; follow the link to the rung for the whole lesson.

  1. c. 385 BC

    Plato, Meno

    Socrates draws squares in the sand and asks a boy how to make a square with twice the area. The boy first guesses wrong, then sees that the answer is the square built on the diagonal. For Plato, geometry shows that the mind can find truths it was never told.

    Go to the rung: Plato: Meno

  2. c. 360 BC

    Plato, Timaeus and Critias

    Plato links the five regular solids to the elements of nature: the cube to earth, the pyramid to fire, and so on. These solids are the only ones with identical regular faces. Geometry, for him, is the plan behind the world.

    Go to the rung: Plato: Timaeus and Critias

  3. c. 300 BC

    Euclid, The Elements, Books I–IV

    Euclid starts with five postulates, such as that a straight line can be drawn between any two points, and proves theorem after theorem. Book I ends with the Pythagorean theorem and its converse. The fifth postulate, about parallel lines, puzzled mathematicians for two thousand years.

    “In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.”

    Euclid, Elements, Book I, Proposition 47. Translated by Thomas L. Heath (Cambridge University Press, 1908). Internet Archive.

    Go to the rung: Euclid: Elements, Books I–IV

  4. c. 300 BC

    Euclid, The Elements, Books XI–XIII

    The last book of the Elements builds the five regular solids inside a sphere and proves there can be no others. Many think this was the goal of the whole work. It joins the geometry of Plato's Timaeus to strict proof.

    “I say next that no other figure, besides the said five figures, can be constructed which is contained by equilateral and equiangular figures equal to one another.”

    Euclid, Elements, Book XIII, after Proposition 18. Translated by Thomas L. Heath (Cambridge University Press, 1908). Internet Archive.

    Go to the rung: Euclid: Elements, Books XI–XIII

  5. 3rd century BC

    Archimedes, On the Equilibrium of Planes, On Floating Bodies, Measurement of a Circle, The Sand-Reckoner, On the Sphere and Cylinder, and others

    Archimedes finds the area of a circle, the volume of a sphere and the area of a parabola's slice, using clever methods of squeezing a shape between figures he could measure. He was proudest of showing a sphere has two-thirds the volume of the cylinder around it.

    Go to the rung: Archimedes: The Works

  6. 1637

    René Descartes, The Geometry

    Descartes shows how to describe curves with equations, so that geometry and algebra can solve each other's problems. Today's graphs on x and y axes grow from his idea. Hard geometric problems become calculations.

    “Any problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction.”

    René Descartes, The Geometry, Book I. Translated by David Eugene Smith and Marcia L. Latham (Open Court, Chicago, 1925). Internet Archive.

    Go to the rung: Descartes: The Geometry

  7. 1902

    Henri Poincaré, Science and Hypothesis

    By Poincaré's time, mathematicians had found consistent geometries in which Euclid's parallel postulate fails. Poincaré argues that asking which geometry is true is like asking whether meters are truer than feet. We choose the one that is most convenient.

    “One geometry cannot be more true than another; it can only be more convenient.”

    Henri Poincaré, Science and Hypothesis, Chapter III, Non-Euclidean Geometries. Translated by W. J. Greenstreet (Walter Scott Publishing, 1905). Project Gutenberg.

    Go to the rung: Poincaré: Science and Hypothesis

  8. 1916 (English 1920)

    Albert Einstein, Relativity: The Special and the General Theory

    Einstein answers that, for physics, the geometry of space-time is something to be found by experiment, and near heavy masses it is not Euclid's. Gravity shows up as curved space-time. Geometry becomes part of physics.

    Go to the rung: Einstein: Relativity

For young readers

Geometry is the math of shapes: lines, circles, triangles and cubes. Euclid, a Greek teacher, showed how to prove facts about shapes step by step, starting from simple rules.

Try this: draw a square, then draw a line from one corner to the opposite corner. Build a new square on that line. It has exactly twice the area of the first one! Socrates showed a boy this trick 2,400 years ago.

Talk about it

  • Why is it not enough to measure a drawing to prove something in geometry?
  • Find shapes around your house: circles, triangles, rectangles, cubes. Which is most common?
  • Can you make a solid with all faces the same regular shape? How many kinds can you find?
  • What would a world look like where parallel lines could meet?

Where else on the site

Every rung tagged with this idea:

These pages are our own short synthesis, written for families. They were inspired by Mortimer J. Adler's Syntopicon, the index of great ideas in volumes 1 and 2 of Great Books of the Western World, but the questions, summaries and groupings are ours, not the Syntopicon's. Short excerpts come from public-domain translations, credited where they appear.

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